Seminar Advanced Topics in Nonlinear Dynamics

Program

Oct 15, 2015

Informal Meeting

Planning

Oct 22, 2015

Sascha Sigmund

Flux sensitivity analysis: A comparism.

In my talk we will compare two different approaches to flux sensitivity analysis, the first one was proposed by Fiedler and Mochizuki and the second one by Shinar and Feinberg. We study especially the flux sensitivity matricies itself and related statements on the matricies.

Phillipo Lappicy

On the zero set of a solution of parabolic equations.

It will be discussed a fundamental result in the theory of scalar parabolic equations: the zero dropping property. The proof of this fact will be given following a well known paper by Angenent.
The talk will be continued on 27.10.2015 in room A6 SR025/026.

Oct 29, 2015

Jia-Yuan

Existence of Rotating Spiral for Complex Ginzburg-Landau Equations on Spheres.

The complex Ginzburg-Landau equation (cGLe) can exhibit spiral patterns. By using a suitable ansatz, the problem of finding rotating spirals for cGLe on spheres is reduced to solving a non-autonomous ODE system with singularities. I will indicate how to solve such ODE system by shooting method and transversality argument.

Nov 5, 2015

Ivan Ovsyannikov(University of Bremen, Germany)

On effect of invisibility of stable orbits in homoclinic bifurcations.

Homoclinic bifurcations are known to give rise to various objects
such as periodic orbits, invariant tori as well as more complicated sets (e.g. strange attractors). An important
role here is played by the so-called saddle value which is the rate of the contraction/expansion of two-dimensional
areas near the saddle fixed point. If the areas are contracted, the bifurcations undergo in the simplest way,
producing stable periodic orbits (exactly one in the flow
case) which are easily observed in one-parametric families. But if 2D areas are expanded, the situation gets
much more complicated. In continuous-time case it was shown by L.P. Shilnikov that unfolding of homoclinic
bifurcations lead to the appearance of infinitely many coexisting periodic orbits (the so-called Shilnikov chaos).
For the discrete-time systems it turns out that stable periodic may be born but they are observed in experiments
with “zero probability”. In my talk I will give the theoretical explanation of this phenomenon.

This is a joint work with S. Gonchenko and D. Turaev (Physica D, 2012).

Nov 12, 2015

Yuya Tokuta

Equilibria of the reaction-diffusion system modeling the convection patterns of Euglena gracilis.

Micro-organisms are known to form spatio-temporal patterns similar to those formed in the Raleigh-Bernard model for thermal convection. Among such micro-organisms, Euglena gracilis form distinct patterns induced by positive/negative photo-taxis and sensitivity to the gradient of light intensity. A model for the convection patterns of Euglena gracilis was proposed by Suematsu et al. and we will discuss equilibria of the system in the simplest case.

Nov 19, 2015

Arne Goedeke

Linear instability of black strings.

Black strings are black hole solutions of Einstein's equations in more than four dimensions. In spacetime dimensions n+1 they have horizon
topology S^{n-1} x S^1. Using numerical simulations, it was discovered
in the 1990s that black strings are linearly unstable. Since then, most
research surrounding this result has focused on understanding physical
aspects of the instability. We will return to the original problem and
present a rigorous proof of the linear instability.

Nov 26, 2015

Bernhard Brehm

Unreachable Heteroclinic Chains in vacuum Bianchi IX.

The Bianchi IX system of ODEs describes the behaviour of a certain class ofhomogeneous anisotropic cosmological models (i.e. solutions to Einstein's equations of general relativity) near the big bang singularity. It is known that there exists an attractor consisting of heteroclinic orbits, which exhibits chaotic behaviour. Previous results from the group [Georgi, Haerterich, Liebscher et al] have shown that certain such heteroclinic chains (forming a countable union of Cantor sets) posess stable manifolds of codimension one and Lipschitz regularity. We will show that only a meagre (Baire-small) set of heteroclinic chains can have a stable object of any meaningful regularity ("connected contracting sets") attached, which especially excludes stable Lipschitz manifolds. If time permits, we will also give an additional justification for the used regularity class of “attached connected contracting sets”. This comes in the form of a comparatively short construction of “attached connected contracting sets” for a large (but still meagre) class of heteroclinic chains.

Dec 03, 2015

Anna Karnauhova

Morse Meanders of Type I and II and the corresponding connection graphs.

Our talk is related to Prof. Bernold Fiedler’s work on sixteen examples of global attractors of one-dimensional parabolic equations. By introducing the right one-shift we will prove under certain assumptions on the arc configurations that we obtain a
class of Morse meanders in the size of the Catalan numbers for the fixed number of arcs. The Morse meanders of this first class will be called of Type I as well as the
associated connection graphs which arise by the blocking and liberalism conditions. Further, it will be possible to deliver necessary and sufficient conditions on connection graphs of Type I for being isomorphic in the graph theoretical sense.
Simultaneously to the first part of the talk, by introducing the left one-shift we will open a second class of meanders which are not Morse meanders. Strictly speaking, our aim will be to argue that it is possible to recover the Morse property of meanders of the secondclass by introducing precisely two maps.
Our study will be accomplished by considering concatenations of both types of Morse meanders and a post-discussion of the non-existence of graph isomorphisms of connection graphs being from two different classes I and II.

We will continue to solve the existence and stability problem of the non-autonomous ODE
system with singularities, which yields spiral patterns of the complex Ginzburg-Landau equation through a spiral ansatz. In this talk,
we will focus on the global bifurcation approach and relate the results with the shooting method.

Dec 17, 2015

Isabelle Schneider

Spatio-temporal feedback control of partial differential equations.

Noninvasive time-delayed feedback control (“Pyragas control”) has been extensively studied and applied in the context of ordinary differential equations. For partial differential equations, almost no results exist up to date. In my talk, I introduce new noninvasive feedback control terms, using both space and time for the control of partial differential equations.

In general relativity, the geometry of the universe near the big bang singularity is tightly related to a discrete map on a circle - the Mixmaster map. This map can be described by an elementary geometric construction involving an equilateral triangle. In this talk, we will consider modified gravity models called Ho\v{r}ava-Lifshitz. In the geometrical construction of the Mixmaster map, this amounts to scaling the equilateral triangle while keeping the rest of the construction unchanged. We will focus on the case where the triangle is larger than for general relativity, which I find the more demanding case mathematically, and probably also the more relevant case physically. As a consequence, overlapping expanding arcs will show up in the discrete dynamics. This fact leads us to model the discrete dynamics by a generalized version of iterated function systems that will be introduced in the talk. Our aim is to prove that the discrete dynamics is chaotic (except in the limiting case where the equilateral triangle is infinitely large).

Jan 14, 2016

Nikita Begun

Dynamics of a Discrete Time System with Stop Operator.

We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by general-equilibrium macroeconomic models with sticky information.

Jan 21, 2016

Bernhard Brehm

Introduction to computability and universal Turing machines.

Jan 28, 2016

Bernhard Brehm

The Bianchi VIII Attractor Theorem and Particle Horizons.

This talk will give a complete overview of my doctoral thesis. The Wainwright-Hsu system of ODE describes the dynamics of spatially homogeneous anisotropic space-times under the vacuum Einstein-Field equations, in the case where the homogeneity is given by either so(3) (Bianchi 9) or sl(2,R) (Bianchi 8). Relevant questions are the open-ended "describe the dynamics!" and the more specific physical "do particle horizons develop?", which boils down to bounding a certain integral. The talk will contain the following parts:
1. Overview: The talk will provide a short overview of the Wainwright-Hsu sytem.
2. Attractor Theorem: The Wainwright-Hsu system contains an invariant set, called the "Mixmaster attractor". Hans Ringstroem proved in 2001 that this set is actually an attractor for Bianchi 9 initial conditions. My thesis proves that this set is is also an attractor for Bianchi 8 initial conditions, and provides a new proof of Ringstroem's result. The talk will give an overview of this proof.
3. Particle horizons: My thesis proves that particle horizons develop for Lebegues a.e. initial condition. The talk will give an overview of this proof.

Feb 04, 2016

Robert Krehl

The Fucik spectrum: Inhomogeneous case

I will give a short introduction to the Fucik spectrum by analyzing the equation -u’’=bu^(+)-au^(-) with Dirichlet boundary conditions. Then, I will try to generalize my approach to prove the existence of a spectrum in the inhomogeneous case. Hopefully, in the end, there will be enough time to indicate some ideas of my current work in progress of finding a representation formula for the spectrum in the inhomogeneous case.

This talks explains the very last part of my Master´s Thesis. Monomolecular reaction networks are modeled by particular directed graphs, which represent reactions as directed arrows. The first part of my thesis, exposed in a talk last semester, was about reformulating a theorem by Fiedler and Mochizuki which characterized, in a graph-language only, the Flux-influence relation. A reaction j´ is flux-influenced by another reaction j* if it responds by a flux variation to perturbations of the reaction rate of j*. This reformulation was done for obtaining a different proof of the transitivity of Flux-influence relation. After a brief recall, we will use this reformulation to derive a characterization of the whole set of arrows which are flux-influenced by a specific one. Nomenclature issues regarding how to call or define new graph-tools for these specific kind of networks are not completely close and suggestions and ideas will be welcome.

Time and Place

Talks usually take place on Thursday at 2:15 p.m.
at the Free University Berlin
Room 130, Arnimallee 3 (rear building), 14195 Berlin.